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Implicit Quantification in N3

From: Dörthe Arndt <dorthe.arndt@ugent.be>
Date: Fri, 16 Jan 2015 12:05:43 +0100
Message-ID: <54B8F087.8060501@ugent.be>
To: public-cwm-talk@w3.org

I am currently trying to write down the basic formal semantics of N3 and 
I had some problems with implicit quantification. Maybe you could help 
me here?

I am not 100% sure how the scope of the universal quantifiers is. If I 
have for example the formula:

     {{?x :p :o}=> {?x :pp :o}}=> {{?x :g :l}=>{?x :k :o}}.

Are the four ?x the same? Does it mean

     1. ∀x: ((p(x,o) → pp(x,o))→(g(x,l)→k(x,o)))


     2. ((∀x_1 p(x_1, o) → pp(x_1, o))→(∀x_2 g(x_2, l)→k(x_2, o)))

I checked in the N3 team submission 
(http://www.w3.org/TeamSubmission/N3/#Quantifica) which states:

"Apart from the set of statements, a formula also has a set of URIs of 
symbols which are universally quantified, and a set of URIs of symbols 
which are existentially quanitified. Variables are then in general 
symbols which have been quantified. There is a also a shorthand syntax 
?x which is the same as :x except that it implies that x is universally 
quantified not in the formula but in its parent formula"

This and the behaviour of the cwm reasoner 
(http://www.w3.org/2000/10/swap/doc/cwm.html) in that case, suggest that 
it is 2. But in the EYE reasoner (http://eulersharp.sourceforge.net/) 
the first option seems to be implemented (and this first interpretation 
would also be easier to formalise).

What about nested formulas? Is the quantification on top level inherited 
or is there a new quantifier for every level? For EYE it is inherited, 
but I tested two examples in cwm:

     1. {?y :p :o}=> {:s :p {?y :pp :o}}.

     2. {:s :p :o}=> {?x :p {?x :pp :o}}.

According to its parsing, the ?y in the first formula are the same, the 
two ?x of the second formula aren't. cwm's interpretation:

     1.  ∀y: (p(y,o) → p(s,pp(y,o))

     2. p(s,o) → (∀x_1: p(x_1,(∀x_2: pp(x_2,o))

So, how is it supposed to be? Maybe I am misunderstanding it, but 
shouldn't the parsing for the two formulas behave similarly?
Why is there a difference?

Thank you in advance for your help!

Kind regards,
Dörthe Arndt

Dörthe Arndt
Researcher Semantic Web
Ghent University - iMinds - Multimedia Lab
Faculty of Engineering and Architecture
Department of Electronics and Information Systems
Gaston Crommenlaan 8 bus 201, B-9050 Ledeberg-Ghent, Belgium

t: +32 9 331 49 59
e: dorthe.arndt@ugent.be
URL:  http://multimedialab.elis.ugent.be
Received on Monday, 19 January 2015 07:09:49 UTC

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